Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $14.3$ years; the standard deviation is $3.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living longer than $23.6$ years.
$14.3$ $11.2$ $17.4$ $8.1$ $20.5$ $5$ $23.6$ $99.7\%$ $0.15\%$ $0.15\%$ We know the lifespans are normally distributed with an average lifespan of $14.3$ years. We know the standard deviation is $3.1$ years, so one standard deviation below the mean is $11.2$ years and one standard deviation above the mean is $17.4$ years. Two standard deviations below the mean is $8.1$ years and two standard deviations above the mean is $20.5$ years. Three standard deviations below the mean is $5$ years and three standard deviations above the mean is $23.6$ years. We are interested in the probability of a seal living longer than $23.6$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. The remaining $0.3\%$ of the seals will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({0.15\%})$ will live less than $5$ years and the other half $({0.15\%})$ will live longer than $23.6$ years. The probability of a particular seal living longer than $23.6$ years is ${0.15\%}$.